Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(5971968\)\(\medspace = 2^{13} \cdot 3^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.1492992.4 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.8.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.0.1492992.4 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 9x^{2} - 12x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 11 + \left(10 a + 27\right)\cdot 41 + \left(40 a + 22\right)\cdot 41^{2} + 13\cdot 41^{3} + \left(9 a + 29\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 7\cdot 41 + 9\cdot 41^{2} + 23\cdot 41^{3} + 40\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 40 a + 14 + \left(30 a + 15\right)\cdot 41 + 10\cdot 41^{2} + \left(40 a + 17\right)\cdot 41^{3} + \left(31 a + 14\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 + 39\cdot 41 + 7\cdot 41^{2} + 10\cdot 41^{3} + 38\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 29 + \left(33 a + 34\right)\cdot 41 + \left(28 a + 9\right)\cdot 41^{2} + \left(3 a + 38\right)\cdot 41^{3} + \left(36 a + 29\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 a + 27 + \left(7 a + 39\right)\cdot 41 + \left(12 a + 21\right)\cdot 41^{2} + \left(37 a + 20\right)\cdot 41^{3} + \left(4 a + 11\right)\cdot 41^{4} +O(41^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(2,5)$ | $-2$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $18$ | $4$ | $(1,2,3,5)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(1,3,4)(2,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.